1. Inferential Statistics
Dr. Mohammed Alahmed
Ph.D. in BioStatistics
(011)

2. Descriptive statistics Inferential statistics Estimation
Hypothesis Testing
Modeling, Predicting

4. Inferential Statistics
Statistical inference is the act of generalizing from a sample to a population with calculated degree of certainty.
but we can only calculate sample statistics
We want to learn about population parameters…
Methods for drawing conclusions about a population from sample data are called

5. Parameters and Statistics
It is essential that we draw distinctions between parameters and statistics

6. Analyse data (e.g. estimate ) to make inferences
Population
(parameters, e.g.,  and )
Sample
collect data from individuals in sample
Data
Analyse data (e.g. estimate ) to make inferences

7. Estimation Point Estimation Interval Estimation
That act of guessing the value of a population parameter
Point Estimation
Estimating a specific value
Interval Estimation
determining the range or interval in which the value of the parameter is thought to be

8. Introduction
The objective of estimation is to determine the approximate value of a population parameter on the basis of a sample statistic.
One of our fundamental questions is: “How well does our sample statistic estimate the value of the population parameter?”
In other word: How close is Sample Statistic to Population Parameter ?

9. Width of confidence interval
A point estimate draws inferences about a population by estimating the value of an unknown parameter using a single value or point.
How much uncertainty is associated with a point estimate of a population parameter?
An interval estimate provides more information about a population characteristic than does a point estimate. It provides a confidence level for the estimate. Such interval estimates are called confidence intervals.
Upper
Confidence
Limit
Lower
Confidence
Limit
Point Estimate
Width of confidence interval

10. A confidence interval for a population characteristic is an interval of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence (the confidence level), the value of the characteristic will be captured inside the interval
Confidence in statistics is a measure of our guaranty that a key value falls within a specified interval.
A Confidence Interval is always accompanied by a probability that defines the risk of being wrong.
This probability of error is usually called α (alpha).

11. Point Estimate ± Margin of Error
How confident do we want to be that the interval estimate contains the population parameter?
The level of confidence (1- α ) is the probability that the interval estimate contains the population parameter.
The most common choices of level of confidence are 0.95, 0.99, and 0.90.
For example: If the level of confidence is 90%, this means that we are 90% confident that the interval contains the population parameter.
The general formula for all confidence intervals is equal to:
Point Estimate ± Margin of Error

12. Choosing a good estimator
Good estimators have certain desirable properties:
Unbiasedness: The sampling distribution for the estimator ‘centers’ around the parameter. (On average, the estimator gives the correct value for the parameter.)
Efficiency: If at the same sample size one unbiased estimator has a smaller sampling error than another unbiased estimator, the first one is more efficient.
Consistency: The value of the estimator gets closer to the parameter as sample size increases. Consistent estimators may be biased, but the bias must become smaller as the sample size increases if the consistency property holds true.

13. Sampling Distribution
Sampling Distribution is a theoretical distribution that shows the frequency of occurrence of values of some statistic computed for all possible samples of size n drawn from some population.
For example, the sampling distribution of 𝑋 is the distribution of values of 𝑥 over all possible samples of size n that could have been selected from the reference population.
A sampling distribution is a distribution of a statistic over all possible samples.
To get a sampling distribution,
1. Take a sample of size N (a given number like 5, 10, or 1000) from a population
2. Compute the statistic (e.g., the mean) and record it.
3. Repeat 1 and 2 a lot (infinitely for large pops).
4. Plot the resulting sampling distribution, a distribution of a statistic over repeated samples

14. Central Limit Theorem
For any population with mean  and standard deviation , the distribution of sample means for sample size n …
will have a mean of 
will have a standard deviation(standard error) of
3. will approach a normal distribution as n gets large ( n ≥ 30 ).

15. Example
Given the population of women has normally distributed weights with a mean of 143 lbs and a standard deviation of 29 lbs,
if one woman is randomly selected, find the probability that her weight is greater than 150 lbs.
if 36 different women are randomly selected, find the probability that their average weight is greater than 150 lbs.

16. Z The probability that her weight is greater than 150 lbs.29
150
 = 143
= 29
Population distribution
0.4052 Z

17. if 36 different women are randomly selected, find the probability that their mean weight is greater than 150 lbs.
150
 = 143
= 4.83
Sampling distribution
0.0735
z = = 1.45
4.83 Z

19. Sampling Error
Sampling error is the error caused by observing a sample instead of the whole population.
Sampling error gives us some idea of the precision of our statistical estimate.
Size of error can be measured in probability samples.
Expressed as “standard error”
of mean, proportion…
Standard error (or precision) depends upon:
size of the sample
distribution of character of interest in population

20. Estimating the population mean 
Estimating the population variance σ2
Estimating the population proportion 

21. Estimating the Population Mean 
A natural point estimator to use for estimating the population mean is the sample mean.
We will use the mean of a sample 𝒙 (statistic) as an estimate of the mean of the population  (parameter).
What properties of 𝒙 make it a desirable estimator of μ? How close is 𝒙 to  ?
Cannot answer question for a particular sample!
Can answer if we can find out about the distribution that describes the variability in the random variable 𝑋 .

22. The (1- α) % confidence interval for μ is given by:
𝒙 ± margin of error (m)
Higher confidence C implies a larger margin of error m (thus less precision in our estimates).
A lower confidence level C produces a smaller margin of error m (thus better precision in our estimates). C z*
−z*
m m

23. Example
The mean height of women age 20 to 30 is normally distributed (bell-shaped) with a mean of 65 inches and a standard deviation of 3 inches.
A random sample of 200 women was taken and the sample mean 𝑥 recorded.
Now IMAGINE taking MANY samples of size 200 from the population of women.
For each sample we record the 𝑥 . What is the sampling distribution of 𝑥 ?

24. We can show that the average of these sample means ( 𝑥 ’s), when taken over a large number of random samples of size n, approximates μ as the number of samples selected becomes large.
In other words, the expected value of 𝑋 over its sampling distribution is equal to μ:
Ε( 𝑋 ) = μ
We refer to 𝑋 as an unbiased estimator of μ. That is:

25. Standard Error of the Mean
We can estimate how much variability there is among potential sample means by calculating the standard error of the mean.
Thus, it is measure of how good our point estimate is likely to be!
The standard error of the mean is the standard deviation of the sampling distribution.

26. In fact: as the sample gets bigger, the sampling distribution…
It is given by:
This equation implies that sampling error decreases as sample size increases.
Thus the larger the sample size, the more precise an estimator 𝑋 is.
In fact: as the sample gets bigger, the sampling distribution…
stays centered at the population mean.
becomes less variable.
becomes more normal.

27. There are 3 different cases:
Interval Estimation of μ when X is normally distributed.
Interval Estimation of μ when X is not normally distributed but the population Standard Deviation is known
Interval Estimation of μ when X is normally distributed and the population Standard Deviation unknown

28. Case 1: Interval Estimation of μ when X is normally distributed
This is the standard situation and you simply use this equation to estimate the population mean at the desired confidence interval:
where Z is the standard normal distribution’s critical value for a probability of α /2 in each tail.

29. μ μl μu Consider a 95% confidence interval: Z Z= -1.96 Z= 1.96 .475
Lower
Confidence
Limit
Upper
.475 Z μ μl μu
Point Estimate

30. When n is greater than 30, this approximation is generally good.
Case 2: Interval Estimation of μ when X is not normally distributed but σ is known
If the distribution of X is not normally distributed, then the central limit theorem can only be applied loosely and we can only say that the sampling distribution is approximately normal.
When n is greater than 30, this approximation is generally good.
Then we use the previous formula to calculate confidence interval.

31. As an estimate for standard error we can use:
Case 3: Interval Estimation of μ when X is normally distributed and σ is unknown
Nearly always  is unknown and is estimated using sample standard deviation s.
If population standard deviation is unknown then it can be shown that sample means from samples of size n are t-distributed with n-1 degrees of freedom.
As an estimate for standard error we can use:

32. Student’s t Distribution
Closely related to the standard normal distribution Z
Symmetric and bell-shaped
Has mean = 0 but has a larger standard deviation
As n increases, the t-dist. approached the Z-dist.
Exact shape depends on a parameter called degrees of freedom (df) which is related to sample size
In this context df = n-1

33. A 100% × (1 − α) CI for the mean μ of a normal distribution with unknown variance is given by:
where is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail

34. Example: Left ventricular ejection fraction (LVEF)
We have 27 patients with acute dilated cardiomyopathy.
1 .19
2 .24
3 .17
4 .40
5 .40
6 .23
7 .20
8 .20
9 .30
10 .19
11 .24
12 .32
13 .32
14 .28
Compute a 95% CI for the true mean LVEF.

35. Use SPSS to do this!
Analyze Descriptive Statistics Explore
= 𝟎.𝟎𝟕𝟗𝟗𝟐 √𝟐𝟕 =𝟎.𝟎𝟏𝟓𝟑𝟖
S =

36. Conclusion:
We are 95% confident that the true mean of LVEF will be contained in the interval ( , )

37. Estimating the population variance σ2
Point Estimation of σ2
Let X1, , Xn be a random sample from some population with mean μ and variance σ2.
The sample variance S 2 is an unbiased estimator of σ2 over all possible random samples of size n that could have been drawn from this population; that is, E (S 2) = σ2.

38. Sampling Distribution of s 2 (Normal Data)
Population variance (s 2) is a fixed (unknown) parameter based on the population of measurements.
Sample variance (s 2) varies from sample to sample (just as sample mean does).
When Y~N (m,s), the distribution of (a multiple of) s 2 is Chi-Square with n -1 degrees of freedom.
(n -1)s 2/s 2 ~ c 2 with df=n -1

39. Chi-Square distributions
Positively skewed with positive density over (0,)
Indexed by its degrees of freedom (df )
Mean=df, Variance=2(df )
Critical Values given in Table 6, p. 824

41. Interval Estimation of σ2
A (1-a)100% Confidence Interval for s 2 (or s) is given by:

43. Example
Assuming an underlying normal distribution, compute 95% CIs for the variance of the Left ventricular ejection fraction (LVEF).
n = 27, s 2 = 0.006, α = 0.05
c2 α/2 = , c2 1−α/2 =

44. Estimation of a population proportion P
We are frequently interested in estimating the proportion of a population with a characteristic of interest, such as:
the proportion of smokers
the proportion of cancer patients who survive at least 5 years
the proportion of new HIV patients who are female

45. The population of interest has a certain characteristics that appears with a certain proportion p. This proportion is a quantity that we usually don’t know, but we would like to estimate it. Since it is a characteristic of the population, we call it a parameter.
To estimate p we draw a representative sample of size n from the population and count the number of individuals (X) in the sample with the characteristic we are looking for. This gives us the estimate X/n.

46. Point estimation of P
This estimate, = X/n, is called the sampling proportion and it is denoted by

47. Sampling Distribution of Proportion
You learned that a binomial can be approximated by the normal distribution if
np  5 and nq  5.
Provided we have a large enough sample size, all the possible sample proportions will be approximately normally distributed with:

48. Confidence Interval for the Population Proportion
An approximate 100% × (1 − α) CI for the binomial parameter p based on the normal approximation to the binomial distribution is given by:

50. Example
Suppose we are interested in estimating the prevalence rate of breast cancer among 50- to 54-year-old women whose mothers have had breast cancer. Suppose that in a random sample of 10,000 such women, 400 are found to have had breast cancer at some point in their lives.
Estimate the prevalence of breast cancer among 50- to 54-year-old women whose mothers have had breast cancer.
Derive a 95% CI for the prevalence rate of breast cancer among 50- to 54-year-old women.

51. Solution:
The best point estimate of the prevalence rate p is given by the sample proportion 𝑝 = =
𝑝 = α = .05 Z1−α/2 = n = 10,000
Therefore, an approximate 95% CI is given by
0.04 ± (.96) = (0.036, 0.044)
What does this mean?

53. Introduction
Researchers often have preconceived ideas about what the values of these parameters might be and wish to test whether the data conform to these ideas.
Used to investigate the validity of a claim about the value of a population characteristic.
For example:
The hospital administrator may want to test the hypothesis that the average length of stay of patients admitted to the hospital is 5 days!
Suppose we want to test the hypothesis that mothers with low socioeconomic status (SES) deliver babies whose birth weights are lower than “normal.”

54. Hypothesis testing is widely used in medicine, dentistry, health care, biology and other fields as a means to draw conclusions about the nature of populations.
Why is hypothesis testing so important? Hypothesis testing provides an objective framework for making decisions using probabilistic methods, rather than relying on subjective impressions.
Hypothesis testing is to provide information in helping to make decisions.

55. General Concepts
A hypothesis is a statement about one or more populations.  There are research hypotheses and statistical hypotheses.
A research hypothesis is the supposition or conjecture that motivates the research.
Statistical hypotheses are stated in such a way that they may be evaluated by appropriate statistical techniques

56. Elements of a statistical hypothesis test
Null hypothesis (H 0): Statement regarding the value(s) of unknown parameter(s). It is the hypothesis to be tested.
Alternative hypothesis (H 1): Statement contradictory to the null hypothesis. It is a statement of what we believe is true if our sample data cause us to reject the null hypothesis.
Test statistic - Quantity based on sample data and null hypothesis used to test between null and alternative hypotheses
Rejection region - Values of the test statistic for which we reject the null in favor of the alternative hypothesis

57. 4 possible outcomes in hypothesis testing
Truth H1 H0
H1 is true and H0 is accepted
Type II error (β)
H0 is true and H0 is accepted
Correct decision
Accept H0
H1 is true and H0 is rejected
Correct decision
H0 is true and H0 is rejected
Type І error (α)
Reject H0
Decision
The probability of a type I error is the probability of rejecting the null hypothesis when H0 is true, denoted by α and is commonly referred to as the significance level of a test.
The probability of a type II error is the probability of accepting the null hypothesis when H1 is true, and usually denoted by β.

58. The general aim in hypothesis testing is to use statistical tests that make α and β as small as possible.
This goal requires compromise because making α small will increase β.
Our general strategy is to fix α at some specific level (for example, .10, .05, .01, . . .) and to use the test that minimizes β or, equivalently, maximizes the power.

59. One-Sample Inferences

60. One-Sample Test for the Mean of a Normal Distribution
One-Sided Tests
Hypothesis:
Null Hypothesis H0 : μ = μ0
Alternative hypothesis H1: μ < μ0 or
H1: μ > μ0
Identify level of significance:
 is a predetermined value (e.g.,  = .05)

61. Test statistic:
If  is known, or if
sample size is large
If  is unknown, or if
sample size is small

62. Critical Value: the rejection region

63. Conclusion: Left-tailed Test: Right-tailed Test
Reject H0 if Z < Z1-α (when use Z - test)
Reject H0 if t < tn-1, α (when use T - test)
Right-tailed Test
Reject H0 if Z > Z1-α (when use Z - test)
Reject H0 if t > t1-α,n-1 (when use T - test)
An Alternative Decision Rule using the p - value Definition

64. P-Value
The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.
The p-value is defined as the smallest value of α for which the null hypothesis can be rejected

65. If the p-value is less than or equal to  ,we reject the null hypothesis (p ≤  )
If the p-value is greater than  ,we do not reject the null hypothesis (p >  )

66. Two-Sided Tests
A two-tailed test is a test in which the values of the parameter being studied (in this case μ) under the alternative hypothesis are allowed to be either greater than or less than the values of the parameter under the null hypothesis (μ0).
To test the hypothesis
H0 : μ = μ0
H1 : μ ≠ μ0

67. Using the Test statistic:
If  is known, or if sample size is large
If  is unknown, or if
sample size is small

68. Decision Rule: Values that differ significantly from H0
Reject H0 if Z >Z1-α/2 or Z< - Z1-α/ (when use Z - test)
Reject H0 if T > t1-α/2,n-1 or t < - t1-α/2,n-1 (when use T- test)
Values that differ significantly from H0

69. Example 1
Among 157 Saudi men ,the mean systolic blood pressure was 146 mm Hg with a standard deviation of 27. We wish to know if on the basis of these data, we may conclude that the mean systolic blood pressure for a population of Saudi is greater than 140. Use  =0.01.

70. Assumption: population is not normal, σ2 is unknown
Data: Variable is systolic blood pressure, n=157 , 𝑋 =146, s=27, α=0.01.
Assumption: population is not normal, σ2 is unknown
Hypotheses: H0 :μ = 140
H1: μ > 140
Test Statistic:
= 2.78

71. What is our final conclusion?
Decision Rule:
we reject H0 if Z > Z1-α
Z0.99= 2.33 from table Z
Decision:
Since 2.78 > 2.33.
then we reject H0
What is our final conclusion?

72. Example 2
The following are the systolic blood pressures of 12 patients undergoing drug therapy for hypertension:
183, 152, 178, 157, 194, 163, 144, 114, 178, 152, 118, 158
Can we conclude on the basis of these data that the population mean of SBP is less than 165? Use  =0.05.
Use SPSS to do this!
Here  is unknown and sample size is small!

73. Analyze Compare Means One-Sample T Test
Solution:
State your hypotheses:
H0 :μ = 165
H1: μ < 165
Analyze Compare Means One-Sample T Test
Select the variable
Set the test value μ0

74. Since p-value > 0.05. Then we fail to reject H0.
The results of this test
Sample mean
Sample standard dev.
T test statistic
Mean from null hyp.
P-value (two-tailed)
Here, the two-tailed P-value is Since we are conducting a one-tailed test, the P-value is 0.315/2 =
Since p-value > Then we fail to reject H0.

75. One-Sample Test for the Population Proportion P (binomial distribution)
Recall that: The sampling distribution of a sample proportion is approximately normal (normal approximation of a binomial distribution) when the sample size is large enough.
Thus, we can easily test the null hypothesis:
H0: p = p0
Alternative Hypotheses:

76. Test Statistic:
Decision Rule:
If H1: P ≠ P0
Reject H 0 if Z > Z1-α/2 or Z < - Z1-α/2
If H1: P > P0
Reject H0 if Z > Z1-α
If H1: P < P0
Reject H0 H0 if Z < - Z1-α

77. Example
Historically, one in five kidney cancer patients survive 5 years past diagnosis, i.e. 20%.
An oncologist using an experimental therapy treats n = 40 kidney cancer patients and 16 of them survive at least 5 years.
Is there evidence that patients receiving the experimental therapy have a higher 5-year survival rate?

78. Determine test criteria
p = the proportion of kidney cancer patients receiving the experimental therapy that survive at least 5 years.
Determine test criteria
Choose a = .05 (may want to consider smaller?) Use large sample test since:
np = (40)(.20) = 8 > 5 and n(1-p) = (40)(.80) = 32 > 5

79. What is our final conclusion?
Z1-α = Z = 1.64
Since Z > Z1-α Reject H0 z
1.64
What is our final conclusion?

80. One-Sample Test for Variance and Standard Deviation
Why test variance ?
In real life, things vary. Even under strictest conditions, results will be different between one another. This causes variances.
When we get a sample of data, apart from testing on the mean to find if it is reasonable, we also test the variance to see whether certain factor has caused greater or smaller variation among the data.
In nature, variation exists. Even identical twins have some differences. This makes life interesting!

81. To draw inference about variability, the parameter of interest is σ2.
Sometimes we are interested in making inference about the variability of processes.
Examples:
The consistency of a production process for quality control purposes.
Investors use variance as a measure of risk.
To draw inference about variability, the parameter of interest is σ2.

82. We know that the sample variance s2 is an unbiased, consistent and efficient point estimator for σ2.
The statistic χ2 has a distribution called Chi-squared, if the population is normally distributed.
The test statistic for testing hypothesis about a population variance is:
χ 2 = 𝒏−𝟏 𝒔𝟐 𝝈𝟎 𝟐
follows a chi-square distribution with degrees of freedom df = n – 1.

83. reject H0 if: χ2 < χ2n-1, α/2 or χ2 > χ2n-1,1- α/2
If we want to test the following hypothesis:
reject H0 if: χ2 < χ2n-1, α/ or χ2 > χ2n-1,1- α/2 χ2
χ2n-1,1- α/2
χ2n-1, α/2
α/2

84. Example
A local balloon company claims that the variance for the time one of its helium balloons will stay afloat is 5 seconds.
A disgruntled customer wants to test this claim.
She randomly selects 23 customers and finds that the variance of the sample is 4.5 seconds.
At  = 0.05, does she have enough evidence to reject the company’s claim?

85. Hypothesis to be tested: Test statistic:
H0 :  2 = 5 (Claim)
H1 :  2  5
Hypothesis to be tested:
Test statistic: X2
36.781
10.982
Fail to reject H0.
Conclusion:
At  = 0.05, there is not enough evidence to reject the claim that the variance of the float time is 5 seconds.
From Chi-square Table

86. Two-Sample Inferences

87. Hypothesis Testing: Two-Sample Inference
A more frequently encountered situation is the two-sample hypothesis-testing problem.
In a two-sample hypothesis-testing problem, the underlying parameters of two different populations, neither of whose values is assumed known, are compared.
Distinguish between Independent and Dependent Sampling.

88. Two samples are said to be independent when the data points in one sample are unrelated to the data points in the second sample.
Two samples are said to be Dependent (paired) when each data point in the first sample is matched and is related to a unique data point in the second sample.
Dependent samples are often referred to as matched-pairs samples.

89. Inference about Two Population Means
The Paired t - test
Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean with  unknown, except that the differences are analyzed.
di = xi2 − xi1
The hypothesis testing problem can thus be considered a one-sample t test based on the differences (di ).

90. To test hypotheses regarding the mean difference of matched-pairs data, the following must be satisfied:
the sample is obtained using simple random sampling
the sample data are matched pairs,
the differences are normally distributed with no outliers or the sample size, n, is large (n ≥ 30).

91. Step 1:
Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways, where μd is the population mean difference of the matched-pairs data.

92. Compute the test statistic which approximately follows
Step 2:
Select a level of significance, , based on the seriousness of making a Type I error.
Step 3:
Compute the test statistic
which approximately follows
Student’ t-distribution with
n-1 degrees of freedom
The values of and sd are the mean and standard deviation of the differenced data.

93. Step 4:
Use Table t to determine the critical value using n-1 degrees of freedom.
Step 5:
Compare the critical value with the test statistic:
If P-value < , reject the null hypothesis.

94. Example: Hypertension
Let’s say we are interested in the relationship between oral contraceptive (OC) use and blood pressure in women.
Identify a group of non-pregnant, premenopausal women of childbearing age (16 – 49 years) who are not currently OC users, and measure their blood pressure, which will be called the baseline blood pressure.
Rescreen these women 1 year later to ascertain a subgroup who have remained non-pregnant throughout the year and have become OC users. This subgroup is the study population.

95. Measure the blood pressure of the study population at the follow-up visit.
Compare the baseline and follow-up blood pressure of the women in the study population to determine the difference between blood pressure levels of women when they were using the pill at follow-up and when they were not using the pill at baseline.
Assume that the SBP of the ith woman is normally distributed at baseline with mean μi and variance σ2 and at follow-up with mean μi + d and variance σ2.

96. If d = 0, then there is no difference between mean baseline and follow-up SBP. If d > 0, then using OC pills is associate with a raised mean SBP. If d < 0, then using OC pills is associated with a lowered mean SBP.
We want to test the hypothesis:
H0: μd = 0 vs. H1: μd ≠ 0.
How should we do this?

97. Solution:
Using SPSS
After Importing your dataset, and providing names to variables, click on:
ANALYZE  COMPARE MEANS  PAIRED SAMPLES T-TEST
For PAIRED VARIABLES, Select the two dependent (response) variables (the analysis will be based on first variable minus second variable)

98. sd
p-value
Since the exact two-sided p-value = .009 < 0.05, we reject the null hypothesis. Therefore, we can conclude that starting OC use is associated with a significant change in blood pressure.

99. A two-sided 100% × (1 − α) CI for the true mean difference (μd) between two paired samples is given by:
Compute a 95% CI for the true increase in mean SBP after starting OCs.
(1.53, 8.07)
Note that the conclusion from the hypothesis test and the confidence interval are the same!

100. Two-Sample T-Test for Independent Samples
Equal Variances:
The independent samples t-test is probably the single most widely used test in statistics.
It is used to compare differences between separate groups.
We want to test the hypothesis:
H0: μ1 = μ2 vs.
H1: μ1 ≠ μ2 or μ1 > μ2 or μ1 < μ2
Sometimes we will be willing to assume that the variance in the two groups is equal:

101. If we know this variance, we can use the z-statistic
Often we have to estimate s2 with the sample variance from each of the samples,
Since we have two estimates of one quantity, we pool the two estimates.

102. The average of and could simply be used as the estimate of σ2 :
The t-statistic based on the pooled variance is very similar to the z-statistic as always:
The t-statistic has a t-distribution with degrees of freedom

103. Constructing a (1-)100 % Confidence Interval for the Difference of Two Means: equal variances
If the two populations are normally distributed or the sample sizes are sufficiently large (n1≥30 and n2≥30), and a (1-)100 % confidence interval about 1 - 2 is given by: [ ]

104. Unequal variance
Often, we are unwilling to assume that the variances are equal!
The test statistic as:
The distribution of this statistic is difficult to derive and we approximate the distribution using a t-distribution with n degrees of freedom
This is called the Satterthwaite or Welch approximation

105. Constructing a (1-)100 % Confidence Interval for the Difference of Two Means (Unequal variance)
If the two populations are normally distributed or the sample sizes are sufficiently large (n1≥30 and n2≥30), a (1-)100 % confidence interval about 1 - 2 is given by: [ ]

106. Equality of variance
Since we can never be sure if the variances are equal, could we test if they are equal?
Of course we can!!
But, remember there is error in every statistical test.
Sometimes it is just preferred to use the unequal variance unless there is a good reason.
Test: H0: s12 = s22 vs. H1: s12 ≠ s22 with significance level  .
To test this hypothesis, we use the sample variances:

107. Testing for the Equality of Two Variances
One way to test if the two variances are equal is to check if the ratio is equal to 1
(H0: ratio=1)
Under the null, the ratio simplifies to
The ratio of 2 chi-square random variables has an F-distribution.
The F-distribution is defined by the numerator and denominator degrees of freedom.
Here we have an F-distribution with n1-1 and n2-1 degrees of freedom

108. Strategy for testing for the equality of means in two independent, normally distributed samples
of two variances
for the equality
Perform F-test
Perform t-test assuming unequal variances
Perform t-test assuming equal variances
Significant
P-value ≤ α
Not significant
P-value > α

109. Example
Use hypothesis testing for Hospital-stay data in Table 2.11 (page 33 in the book) to answer (2.3).
Use SPSS to do the analysis!
2.3: It is of clinical interest to know if the duration of hospitalization is affected by whether a patient has received antibiotics.
That is to compare the mean duration of hospitalization between antibiotic users and non-antibiotic users.

110. Solution
H0 : μ1 = μ2
H1 : μ1 ≠ μ2
μ1 the mean duration of hospitalization for antibiotic users
μ2 the mean duration of hospitalization for non-antibiotic users

111. SPSS output H0 : μ1 - μ2 = 0 vs. H1 : μ1 - μ2 ≠ 0.
H0: s12 = s22 vs. H1: s12 ≠ s22
H0 : μ1 - μ2 = 0 vs.
H1 : μ1 - μ2 ≠ 0.

112. Conclusion
First step in comparing the two means is to perform the F-test for the equality of two variances in order to decide whether to use the t-test with equal or unequal variances.
The p -value = > Thus the variances DO NOT differ significantly, and a two-sample t-test with equal variances should be used.
Therefore, refer to the Equal Variance row, where the t-statistic is 1.68 with degrees of freedom d’(df). = 23.
The corresponding two-tailed p-value = > 0.05.
Thus no significant difference exists between the mean duration of hospitalization in these two groups.

113. The Treatment of Outliers
Outliers can have an important impact on the conclusions of a study.
It is important to definitely identify outliers and either:
Exclude them! ( NOT recommended) or
Perform alternative analyses with and without the outliers present.
Use a method of analysis that minimizes their effect on the overall results.

114. Box-Plot

115. Inference about Two Population Proportions
Testing hypothesis about two population proportion (P1 , P2 ) is carried out in much the same way as for difference between two means when condition is necessary for using normal curve are met.

116. We have the following:
Data: sample size (n1 وn2),
- Characteristic in two samples (x1 , x2),
- sample proportions
- Pooled proportion

117. Assumption: Two populations are independent
Assumption: Two populations are independent. Hypotheses: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways
Select a level of significance α, based on the seriousness of making a Type I error.

118. Compute the test statistic:
Decision Rule:
Compare the critical value with the test statistic:
Or, if P-value < α , reject the null hypothesis

119. Example
An economist believes that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access.
He obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains another random sample of 750 rural households and finds that 292 of them have Internet access.
Test the economist’s claim at the  =0.05 level of significance.

120. Solution:
The samples are simple random samples that were obtained independently.
x1=338, n1=800, x2=292 and n2=750, so
We want to determine whether the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access.

121. H0: p1 - p2=0 versus H1: p1 - p2 > 0
So, we test the hypothesis:
H0: p1 = p2 versus H1: p1 > p2
or, equivalently,
H0: p1 - p2=0 versus H1: p1 - p2 > 0
The level of significance is  = 0.05

122. The pooled estimate:
The test statistic is:

123. This is a right-tailed test with  =0.05
The critical value is Z0.05=1.645
Since the test statistic, z0=1.33 is less than the critical value z.05=1.645,
Then we fail to reject the null hypothesis.
Conclusion:
The percentage of urban households with Internet access is the same as the percentage of rural households with Internet access.

124. Constructing a (1-)100% Confidence Interval for the Difference between Two Population Proportions
To construct a (1-)100% confidence interval for the difference between two population proportions:

125. The Relationship Between Hypothesis Testing and Confidence Intervals
(Two-Sided Case)
Suppose we are testing
H0: μ = μ0 vs.
H1: μ ≠ μ0
H0 is rejected with a two-sided level  test if and only if the two-sided 100% (1 −  ) CI for μ does not contain μ0
H0 is accepted with a two-sided level  test if and only if the two-sided 100% (1 −  ) CI for μ does contain μ0

126. Is the Population Normal?
Is n ≥ 30 ?
Use non-parametrics
test
Use z-test
Is σ known?
Use t-test No
Yes
Yes No No
Yes